In our setting with paired music conditioning \(\mathbf {c}\) , we reverse the forward diffusion process by learning to estimate \(\mathbf {\hat{x}}_\theta (\mathbf {z}_t, t, \mathbf {c}) \approx \mathbf {x}\)  with model parameters \(\theta \)  for all \(t\) .
We optimize \(\theta \)  with the “simple” objective introduced in Ho et al. [1]}:
\( {\mathcal {L}_\text{simple} = \mathbb {E}_{\mathbf {x}, t}\left[\Vert  \mathbf {x} - \mathbf {\hat{x}}_\theta (\mathbf {z}_t, t, \mathbf {c})\Vert _2^2\right].}\) 
